Properties of Real Numbers

1
Properties of Real Numbers

• Unit 2, Lesson 1
• Online Algebra 1
• VHS_at_pwcs
• Cami Craig

2
Properties

• In this lesson we are going to look at
  properties, rules of mathematics that can be
  proven.
• We will be looking at properties of equality, and
  real number properties.

3
Properties of Real Numbers

• Properties of Real Numbers include
• Commutative Property
• Associative Property
• Identity Property
• Distributive Property
• Inverse Property
• Closure Property

4
Commutative Properties

• Commutative Properties deal with order. Order in
  multiplication and addition do not matter!

Addition a b b a Or 5 9 9 5 ..Is
this true? Try it! Of course it is 14 14
Multiplication ab ba Or 5(9) 9(5) ..Is this
true? Try it! Of course it is 45 45
5
Commutative Properties and Subtraction

• Does the Commutative Property hold for
  subtraction and division?
• Lets try

If the commutative property holds for subtraction
then the following should be true 6 3 3
6 But we know it isnt true 6 3 3 and 3 6
-3 So the commutative property does NOT work for
subtraction.
6
Commutative Properties and Division

• Does the Commutative Property hold for division?
• Lets try

If the commutative property holds for division
then the following should be true 10 5 5
10 But we know it isnt true 10 5 2 and 5
10 0.5 So the commutative property does NOT
work for division.
7
Associative Properties

• Associative Properties deal with Grouping.
  Grouping in multiplication and addition do not
  matter! But just like the commutative properties
  the associative property does not apply to
  subtraction.

Multiplication a(bc) (ab)c Or 4 x (6 x 2) (4
x 6) x 2 ..Is this true? Try it! 4 x (6 x 2)
(4 x 6) x 2 4 x 12 24 x 2 48 48
Addition a (b c) (a b) c Or 11 (5 9)
(11 5) 9 ..Is this true? Try it! 11 (
5 9) (11 5) 9 11 14 16 9 25 25
8
Associative and Commutative Properties

• How can you tell these properties apart?

Commutative Properties 8 11 11 8 4 x 5 5
x 4
Associative Properties 3 ( 6 4) (3 6)
4 7 x (3 x 5) (7 x 3) x 5
Notice that in the commutative property the order
of the numbers change, while in the associative
properties the order stays the same, but the
grouping changes.
9
Distributive Property

• The distributive property is
• a(b c) ab ac
• Or
• 2( 4 5) 2 x 4 2 x 5
• I like to call the Distributive Property the fair
  share property, because the number on the outside
  of the parentheses is multiplied to both numbers
  in the parentheses.

10
Identity Properties

• The Identity Properties deal with getting back
  the same thing.

Addition When we add 0 to a number, we get that
original number back For example A 0 A -4
0 -4 We actually call 0 the Additive Identity
Element.
Multiplication When we multiply 1 to a number, we
get that original number back For example 1a
a -15(1) -15 We actually call 1 the
Multiplicative Identity Element.
11
Inverse and Closure Properties

• The inverse property for addition states that a
  -a 0.
• The inverse property for multiplication states
  that a x 1/a 1.
• The closure property for addition states that a
  b is a real number.
• The closure property for multiplication states
  that a x b is a real number.

12
Lets see what you have learned so far!What
property does each example represent?

1. -2(3 4) -2 x 3 -2 x 4
2. 5 (3 6) (3 6) 5
3. 5(1) 5
4. 17 x (8 x 2) (17 x 8) x 2
5. 9 0 9
6. 4 x ¼ 1

• Distributive Property
• Commutative Property
• Notice that although there are parentheses, it is
  the order that changes not the grouping.
• Identity Property of Multiplication
• Associative Property
• Identity Property of Addition
• Inverse Property of Mult.

13
Properties of Equality

• Properties of equality include the following
• The Reflexive Property
• The Symmetric Property
• The Transitive Property

14
Properties of Equality

• Reflexive Property of Equality
• a a
• -9 -9
• Symmetric Property of Equality
• If a b, then b a.
• If 15x 45, then 45 15x
• Transitive Property of Equality
• If a b and b c, then a c
• If d 3y and 3y 6, then d 6.

15
ReviewWhat property is each of the following an
example of?

• 9 9
• a 8 8 a
• If x 8 9, and 9 4 5, then x 8 4 5
• 3(x 7) 3x 21
• 5 x 1 5
• If 16 4x, then 4x 16
• 7 -7 0

1. Reflexive
2. Commutative
3. Transitive
4. Distributive
5. Identity
6. Symmetric
7. Inverse